Discrete Mathematics. Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y ∈...
Discrete Mathematics.
Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y
∈ A,xRy ↔ x|y.
(a) Is R antisymmetric? Prove, or give a counterexample.
(b) Draw the Hasse diagram for R.
(c) Find the greatest, least, maximal, and minimal elements of R
(if they exist).
(d) Find a topological sorting for R that is different from the ≤ relation.
Homework Answers
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Discrete Mathematics.
Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y
∈...
Discrete Mathematics. Let A = {2,4,6,8,10}, and define a relation R on A as ∀x,y ∈...
Discrete Mathematics. Let A = {2,4,6,8,10}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ 4|(x−y). (a) Show R is an equivalence relation. (b) Give R explicitly in terms of its elements. (c) Draw the directed graph of R. (d) List all the distinct equivalence classes of R.
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw...
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw the Hasse diagram for . b) Find all maximal and minimal elements. c) Find lub({6,12}). a) Find glb({6,12}). e) What is the least element? The greatest element? Q-4. [8+3+3+3+3 marks] Let R be the partial order relation defined on A = {2,3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x|y. a) Draw the Hasse diagram for R. b) Find all maximal...
13 pts) Let R be the relation on R deÖned by xRy means "sin2 (x) +...
13 pts) Let R be the relation on R deÖned by
xRy means "sin2 (x) + cos2 (y) = 1".
Recall the Pythagorean identity: 8u 2 R we have sin2 (u) +
cos2 (u) = 1.
(a) (9 pts) PROVE that R is an equivalence relation on
R.
(b) (4 pts) Describe all elements of the (inÖnite) equivalence
class [0].
Recall: sin(0) = 0 and cos(0) = 1.
2. (13 pts) Let R be the relation on R defined by...
Let R be the relation on N defined by xRy iff 2 divides x+y. R is...
Let R be the relation on N defined by xRy iff 2 divides x+y. R is an equivalence relation. You do not have to prove that R is an equivalence relation. True or False: 3 ∈ 4/R.
Show your work, please 4. Partial Orders Let P be the collection of all subsets of...
Show your work, please
4. Partial Orders Let P be the collection of all subsets of X = {a,b,c,d} that have at least two elements. (So {a,c} € P, but {b} P.) Consider the subset relation C as a partial order on P. For example, {a,b} = {a,b,c}. Draw the Hasse diagram, and find any maximum/minimum elements, and maximal/minimal elements.
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transi...
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transitivity). Be sure to give a short explanatio if the property holds and a specific counterexample if it does not hold. b) Is this a Partial Ordering? Remember to check all three criteria (Reflexivity Transitivity, and Anti-Symmetry). Be sure to give a short explanation if...
(e) Define a relation R on Z as xRy if and only if m|(x - y)....
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
Let the relation R be defined on the set {x ∈ R | 0 ≤ x ≤ 1} by xRy ⇔ ∃t(x + t = y and 0 ≤ t ≤ 1)...
Let the relation R be defined on the set {x ∈ R | 0 ≤ x ≤ 1} by xRy ⇔ ∃t(x + t = y and 0 ≤ t ≤ 1) Is R transitive?
Discrete Mathematics 22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1. What is the transitive closure of r? 23. Write an algorithm in pseudo code that converts nu...
Discrete Mathematics
22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1. What is the transitive closure of r? 23. Write an algorithm in pseudo code that converts numbers in decimal representation to octal (base 8) representation 24. Prove that the set of integers in countable
22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1....
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw the Hasse diagram for . b) Find all maximal and minimal elements. c) Find lub({6,12}). a) Find glb({6,12}). e) What is the least element? The greatest element? Q-4. [8+3+3+3+3 marks] Let R be the partial order relation defined on A = {2,3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x|y. a) Draw the Hasse diagram for R. b) Find all maximal...
13 pts) Let R be the relation on R deÖned by
xRy means "sin2 (x) + cos2 (y) = 1".
Recall the Pythagorean identity: 8u 2 R we have sin2 (u) +
cos2 (u) = 1.
(a) (9 pts) PROVE that R is an equivalence relation on
R.
(b) (4 pts) Describe all elements of the (inÖnite) equivalence
class [0].
Recall: sin(0) = 0 and cos(0) = 1.
2. (13 pts) Let R be the relation on R defined by...
Show your work, please
4. Partial Orders Let P be the collection of all subsets of X = {a,b,c,d} that have at least two elements. (So {a,c} € P, but {b} P.) Consider the subset relation C as a partial order on P. For example, {a,b} = {a,b,c}. Draw the Hasse diagram, and find any maximum/minimum elements, and maximal/minimal elements.
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transitivity). Be sure to give a short explanatio if the property holds and a specific counterexample if it does not hold. b) Is this a Partial Ordering? Remember to check all three criteria (Reflexivity Transitivity, and Anti-Symmetry). Be sure to give a short explanation if...
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
Discrete Mathematics
22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1. What is the transitive closure of r? 23. Write an algorithm in pseudo code that converts numbers in decimal representation to octal (base 8) representation 24. Prove that the set of integers in countable
22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1....
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
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